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Theorem pnt 22862
Description: The Prime Number Theorem: the number of prime numbers less than  x tends asymptotically to  x  /  log (
x ) as  x goes to infinity. This is Metamath 100 proof #5. (Contributed by Mario Carneiro, 1-Jun-2016.)
Assertion
Ref Expression
pnt  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1

Proof of Theorem pnt
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1re 9384 . . . . . . 7  |-  1  e.  RR
21rexri 9435 . . . . . 6  |-  1  e.  RR*
3 1lt2 10487 . . . . . 6  |-  1  <  2
4 df-ioo 11303 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
5 df-ico 11305 . . . . . . 7  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
6 xrltletr 11130 . . . . . . 7  |-  ( ( 1  e.  RR*  /\  2  e.  RR*  /\  w  e. 
RR* )  ->  (
( 1  <  2  /\  2  <_  w )  ->  1  <  w
) )
74, 5, 6ixxss1 11317 . . . . . 6  |-  ( ( 1  e.  RR*  /\  1  <  2 )  ->  (
2 [,) +oo )  C_  ( 1 (,) +oo ) )
82, 3, 7mp2an 672 . . . . 5  |-  ( 2 [,) +oo )  C_  ( 1 (,) +oo )
9 resmpt 5155 . . . . 5  |-  ( ( 2 [,) +oo )  C_  ( 1 (,) +oo )  ->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) ) )
108, 9mp1i 12 . . . 4  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) )
118sseli 3351 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  ( 1 (,) +oo ) )
12 ioossre 11356 . . . . . . . . . . 11  |-  ( 1 (,) +oo )  C_  RR
1312sseli 3351 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  x  e.  RR )
1411, 13syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR )
15 2re 10390 . . . . . . . . . . 11  |-  2  e.  RR
16 pnfxr 11091 . . . . . . . . . . 11  |- +oo  e.  RR*
17 elico2 11358 . . . . . . . . . . 11  |-  ( ( 2  e.  RR  /\ +oo  e.  RR* )  ->  (
x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  < +oo ) ) )
1815, 16, 17mp2an 672 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  < +oo ) )
1918simp2bi 1004 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  2  <_  x )
20 chtrpcl 22512 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
2114, 19, 20syl2anc 661 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  RR+ )
22 0red 9386 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  0  e.  RR )
231a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  1  e.  RR )
24 0lt1 9861 . . . . . . . . . . . 12  |-  0  <  1
2524a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  0  <  1 )
26 eliooord 11354 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) +oo )  ->  ( 1  <  x  /\  x  < +oo ) )
2726simpld 459 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  1  < 
x )
2822, 23, 13, 25, 27lttrd 9531 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  0  < 
x )
2913, 28elrpd 11024 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) +oo )  ->  x  e.  RR+ )
3011, 29syl 16 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
3121, 30rpdivcld 11043 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  /  x )  e.  RR+ )
3231adantl 466 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  x )  e.  RR+ )
33 ppinncl 22511 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
3414, 19, 33syl2anc 661 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  NN )
3534nnrpd 11025 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  RR+ )
3613, 27rplogcld 22077 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  ( log `  x )  e.  RR+ )
3711, 36syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  RR+ )
3835, 37rpmulcld 11042 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  RR+ )
3921, 38rpdivcld 11043 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  e.  RR+ )
4039adantl 466 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
4130ssriv 3359 . . . . . . . 8  |-  ( 2 [,) +oo )  C_  RR+
42 resmpt 5155 . . . . . . . 8  |-  ( ( 2 [,) +oo )  C_  RR+  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x ) ) )
4341, 42ax-mp 5 . . . . . . 7  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x ) )
44 pnt2 22861 . . . . . . . 8  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1
45 rlimres 13035 . . . . . . . 8  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  ~~> r  1 )
4644, 45mp1i 12 . . . . . . 7  |-  ( T. 
->  ( ( x  e.  RR+  |->  ( ( theta `  x )  /  x
) )  |`  (
2 [,) +oo )
)  ~~> r  1 )
4743, 46syl5eqbrr 4325 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  x ) )  ~~> r  1 )
48 chtppilim 22723 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
4948a1i 11 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
50 ax-1ne0 9350 . . . . . . 7  |-  1  =/=  0
5150a1i 11 . . . . . 6  |-  ( T. 
->  1  =/=  0
)
5239rpne0d 11031 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  =/=  0 )
5352adantl 466 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
5432, 40, 47, 49, 51, 53rlimdiv 13122 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( theta `  x )  /  x
)  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) )  ~~> r  ( 1  /  1 ) )
5514recnd 9411 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  CC )
56 chtcl 22446 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
5713, 56syl 16 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) +oo )  ->  ( theta `  x )  e.  RR )
5857recnd 9411 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  ( theta `  x )  e.  CC )
5911, 58syl 16 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  CC )
6055, 59mulcomd 9406 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  x.  ( theta `  x
) )  =  ( ( theta `  x )  x.  x ) )
6160oveq2d 6106 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) )  =  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( ( theta `  x
)  x.  x ) ) )
6238rpcnd 11028 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  CC )
6330rpne0d 11031 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  =/=  0 )
6421rpne0d 11031 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  =/=  0
)
6562, 55, 59, 63, 64divcan5d 10132 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( ( theta `  x
)  x.  x ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
6661, 65eqtrd 2474 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
6738rpne0d 11031 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  =/=  0 )
6859, 55, 59, 62, 63, 67, 64divdivdivd 10153 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) )  =  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) ) )
6934nncnd 10337 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  CC )
7037rpcnd 11028 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  CC )
7137rpne0d 11031 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  =/=  0
)
7269, 55, 70, 63, 71divdiv2d 10138 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  x ) )
7366, 68, 723eqtr4d 2484 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) )  =  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
7473mpteq2ia 4373 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )
75 1div1e1 10023 . . . . 5  |-  ( 1  /  1 )  =  1
7654, 74, 753brtr3g 4322 . . . 4  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
7710, 76eqbrtrd 4311 . . 3  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,) +oo ) )  ~~> r  1 )
78 ppicl 22468 . . . . . . . . . 10  |-  ( x  e.  RR  ->  (π `  x )  e.  NN0 )
7913, 78syl 16 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) +oo )  ->  (π `  x
)  e.  NN0 )
8079nn0red 10636 . . . . . . . 8  |-  ( x  e.  ( 1 (,) +oo )  ->  (π `  x
)  e.  RR )
8129, 36rpdivcld 11043 . . . . . . . 8  |-  ( x  e.  ( 1 (,) +oo )  ->  ( x  /  ( log `  x
) )  e.  RR+ )
8280, 81rerpdivcld 11053 . . . . . . 7  |-  ( x  e.  ( 1 (,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  e.  RR )
8382recnd 9411 . . . . . 6  |-  ( x  e.  ( 1 (,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  e.  CC )
8483adantl 466 . . . . 5  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  CC )
85 eqid 2442 . . . . 5  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  =  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
8684, 85fmptd 5866 . . . 4  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) : ( 1 (,) +oo ) --> CC )
8712a1i 11 . . . 4  |-  ( T. 
->  ( 1 (,) +oo )  C_  RR )
8815a1i 11 . . . 4  |-  ( T. 
->  2  e.  RR )
8986, 87, 88rlimresb 13042 . . 3  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  ~~> r  1  <->  ( (
x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  |`  (
2 [,) +oo )
)  ~~> r  1 ) )
9077, 89mpbird 232 . 2  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
9190trud 1378 1  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 965    = wceq 1369   T. wtru 1370    e. wcel 1756    =/= wne 2605    C_ wss 3327   class class class wbr 4291    e. cmpt 4349    |` cres 4841   ` cfv 5417  (class class class)co 6090   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282    x. cmul 9286   +oocpnf 9414   RR*cxr 9416    < clt 9417    <_ cle 9418    / cdiv 9992   NNcn 10321   2c2 10370   NN0cn0 10578   RR+crp 10990   (,)cioo 11299   [,)cico 11301    ~~> r crli 12962   logclog 22005   thetaccht 22427  πcppi 22430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-disj 4262  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-fi 7660  df-sup 7690  df-oi 7723  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ioo 11303  df-ioc 11304  df-ico 11305  df-icc 11306  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-fac 12051  df-bc 12078  df-hash 12103  df-shft 12555  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-limsup 12948  df-clim 12965  df-rlim 12966  df-o1 12967  df-lo1 12968  df-sum 13163  df-ef 13352  df-e 13353  df-sin 13354  df-cos 13355  df-pi 13357  df-dvds 13535  df-gcd 13690  df-prm 13763  df-pc 13903  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-hom 14261  df-cco 14262  df-rest 14360  df-topn 14361  df-0g 14379  df-gsum 14380  df-topgen 14381  df-pt 14382  df-prds 14385  df-xrs 14439  df-qtop 14444  df-imas 14445  df-xps 14447  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-submnd 15464  df-mulg 15547  df-cntz 15834  df-cmn 16278  df-psmet 17808  df-xmet 17809  df-met 17810  df-bl 17811  df-mopn 17812  df-fbas 17813  df-fg 17814  df-cnfld 17818  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-cld 18622  df-ntr 18623  df-cls 18624  df-nei 18701  df-lp 18739  df-perf 18740  df-cn 18830  df-cnp 18831  df-haus 18918  df-cmp 18989  df-tx 19134  df-hmeo 19327  df-fil 19418  df-fm 19510  df-flim 19511  df-flf 19512  df-xms 19894  df-ms 19895  df-tms 19896  df-cncf 20453  df-limc 21340  df-dv 21341  df-log 22007  df-cxp 22008  df-em 22385  df-cht 22433  df-vma 22434  df-chp 22435  df-ppi 22436  df-mu 22437
This theorem is referenced by: (None)
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